defines a bounded operator in \(L_2(\Gamma ):=L_2(\mathcal {H}^m|_{\Gamma })\) . This fundamental result was first obtained by Calderon in [1]} for 1–dimensional Lipschitz graphs in the complex plane with sufficiently small Lipschitz constant. (In this case the 1–dimensional Riesz kernel \(R_1\)  essentially coincides with the Cauchy kernel \(k(z)=z^{-1}, z \in .) The restriction on the Lipschitz constant was removed a few years later by Coifman, McIntosh and Meyer \cite {cmm}. Finally, Coifman, David and Meyer \cite {cdm} proved that \) Tm\( is bounded in \) L2()\( for all \) m\(--dimensional Lipschitz graphs \)\( by showing that the \) m\(--dimensional case can be reduced to the \) 1\(--dimensional case via the method of rotations.\) The \(L_2\) –boundedness of Riesz transforms on Lipschitz graphs has been pivotal for the research program which started in the early 80s with the aim of relating the analytic behavior of singular integrals on subsets of \(\mathbb {R}^n\)  to the geometric structure of these sets. In particular, David and Semmes [2]}, [3]} developed the theory of uniform rectifiability hoping to characterize the \(m\) –Ahlfors regular sets \(E \subset \mathbb {R}^n\)  on which the Riesz transforms \(T^m, m \in (0,n),\)  are bounded in \(L_2(E)\) ; uniformly rectifiable sets can be built out of Lipschitz graphs and can be approximated by Lipschitz graphs at most locations and scales. David proved in [4]} that if \(E\)  is \(m\) –uniformly rectifiable then \(T^m\)  is bounded in \(L^2(E)\) . He and Semmes [3]} conjectured that the converse is also true. That is, if \(E\)  is an \(m\) –Ahlfors regular set such \(T^m\)  is bounded in \(L^2(E)\)  then \(E\)  is \(m\) –uniformly rectifiable. The conjecture was proved by Mattila, Melnikov and Verdera in [6]} for \(m=1\)  and by Nazarov, Tolsa and Volberg [7]} for \(m=n-1\) . It remains open for integers \(m \in (1,n-1)\) .
